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Reactors
Reactors are used in electronic power equipment to smooth out ripple voltage in dc supplies, so they carry direct current in the coils. It is common practice to build such reactors with air gaps in the core to prevent dc saturation. The air gap, size of the core, and number of turns depend upon three interrelated factors: inductance desired; direct current in the winding; and ac volts across the winding.
Magnetic flux through the coil has two component lengths of path: the air gap l_{g}, and the length of the core l_{c}. The core length l_{c} is much greater geometrically than the air gap l_{g}, as indicated in Fig. 57, but the two components do not add directly because their permeabilities are different. In the air gap, the permeability is unity, whereas in the core its value depends on the degree of saturation of the iron. The effective length of the magnetic path is l_{g} + (l_{c}/μ), where μ is the permeability for the steady or dc component of flux. Reactor design is, to a large extent, the proportioning of values of air gap and magnetic path length divided by permeability. If the air gap is relatively large, the reactor inductance is not much affected by changes in μ; it is then called a linear reactor. If the air gap is small, changes in μ due to current or voltage variations cause inductance to vary; then the reactor is nonlinear. When direct current flows in an ironcore reactor, a fixed magnetizing force H_{dc} is maintained in the core. This is shown in Fig. 64 as the vertical line H_{dc} to the right of zero H in a typical ac hysteresis loop, the upper half DB_{m}D' of which corresponds to that in Fig. 21. Increment ΔH of ac magnetization, superposed on H_{dc}, causes flux density increment ΔB, with permeability μ_{Δ} equal to the slope of dotted line AB_{m}. ΔB is twice the peak ac induction B_{ac}. It will be recalled from Fig. 19 that the normal induction curve OB_{m} is the locus of the end points of a series of successively smaller major hysteresis loops. Since the top of the minor loop always follows the left side of a major loop, as H_{dc} is reduced in successive steps the upper ends of corresponding minor loops terminate on the normal induction curve. Dottedline slopes of a series of minor loops are shown in Fig. 64, the midpoints of which are C, C', C", and C'".
Increment of induction ΔB is the same for each minor loop. It will be seen that the width of the loop ΔH is smaller, and hence ma is greater, as H_{dc} is made smaller. Midpoints C, C', etc., form the locus of dc induction. The slope of straight line OC is the dc permeability for core magnetization H_{dc}. It is much greater than the slope of AB_{m}. Hence incremental permeability is much smaller than dc permeability. This is true in varying degree for all the minor loops. The smaller ΔB is, the less the slope of a minor loop becomes, and consequently the smaller the value of incremental permeability μ_{Δ}. The curve in Fig. 65 marked μ is the normal permeability of 4% silicon steel for steady values of flux, in other words, for the dc flux in the core.
It is 4 to 20 times as great as the incremental permeability μ_{Δ} for a small alternating flux superposed upon the dc flux. The ratio of μ to μ_{Δ} gradually increases as dc flux density increases. Because of the low value of μ_{Δ} for minute alternating voltages, the effective length of magnetic path l_{g} + (I_{c}/μ_{Δ}) is considerably greater for alternating than for steady flux. But the inductance varies inversely as the length of ac flux path. If, therefore, the incremental permeability is small enough to make l_{c}/μ_{Δ}large compared to l_{g}, it follows that small variations in l_{g} do not affect the inductance much. For this reason the exact value of the air gap is not important with small alternating voltages. Reactor size, with a given voltage and ratio of inductance to resistance, is proportional to the stored energy LI^{2}. For the design of reactors carrying direct current, that is, the selection of the right number of turns, air gap, and so on, a simple method was originated by C. R. Hanna.^{(1)} By this method, magnetic data are reduced to curves such as Fig. 66, plotted between LI^{2}/V and NI/l_{c} from which reactors can be designed directly. The various symbols in the coordinates are: L = ac inductance in henrys A_{c} = cross section of core in square inches I = direct current in amperes V = volume of iron core in cubic inches = A_{c}l_{c} (see Fig. 57 for core dimensions) l_{c} = length of core in inches N = number of turns in winding l_{g} = air gap in inches
Each curve of Fig. 66 is the envelope of a family of fixed airgap curves such as those shown in Fig. 67.
These curves are plots of data based upon a constant small ac flux (10 gauss) in the core but a large and variable dc flux. Each curve has a region of optimum usefulness, beyond which saturation sets in and its place is taken by a succeeding curve having a larger air gap. A curve tangent to the series of fixed airgap curves is plotted as in Fig. 66, and the regions of optimum usefulness are indicated by the scale l_{g}/l_{c}. Hence Fig. 66 is determined mainly by the dc flux conditions in the core and represents the most LI^{2} for a given amount of material. Figure 67 illustrates how the exact value of air gap is of little consequence in the final result. The dotted curve connecting B and C is for a 6mil gap. Point Y' represents the maximum inductance that could be obtained from a given core for NI/l_{c} = 19. Point Y is the inductance obtained if a gap of either 4 or 8 mils is used. The difference in inductance between Y and Y' is 4 per cent, for a difference in air gap of 33 per cent. An example will show how easy it is to make a reactor according to this method.


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